Periodica Mathematica Hungarica

, Volume 55, Issue 1, pp 1–9 | Cite as

A characterization of some graph classes using excluded minors

  • Janka ChlebíkováEmail author


In this article we present a structural characterization of graphs without K 5 and the octahedron as a minor. We introduce semiplanar graphs as arbitrary sums of planar graphs, and give their characterization in terms of excluded minors. Some other excluded minor theorems for 3-connected minors are shown.

Key words and phrases

excluded minors k-sum of graphs tree-width of a graph 

Mathematics subject classification number

05C75 05C83 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Arnborg, A. Proskurowski and D. Corneil, Forbidden minors characterization of partial 3-trees, Discrete Math., 80 (1990), 1–19.zbMATHCrossRefGoogle Scholar
  2. [2]
    J. Chlebíková, Contributions to Algorithmic Graph Theory: Treewidth, Minors, and Balanced Partitions, Ph.D. thesis, Faculty of Mathematics and Physics, Comenius University, Bratislava, January 2000.Google Scholar
  3. [3]
    R. Diestel, Graph decompositions. A study in infinite graph theory, The Clarendon Press, Oxford University Press, New York, 1990.zbMATHGoogle Scholar
  4. [4]
    G. A. Dirac, Some results concerning the structure of graphs, Canad. Math. Bull., 6 (1963), 183–210.zbMATHGoogle Scholar
  5. [5]
    J. Halin, Über einen graphentheoretischen Basisbegriff und seine Anwendung auf Färbungsprobleme, Dissertation, Köln, 1962.Google Scholar
  6. [6]
    J. Halin and H. A. Jung, Über Minimalstrukturen von Graphen, insbesondere von n-fach zusammengehägenden Graphen, Math. Ann., 152 (1963), 75–94.zbMATHCrossRefGoogle Scholar
  7. [7]
    J. Maharry, An Excluded Minor Theorem for the Octahedron, J. Graph Theory, 31 (1999), 95–100.zbMATHCrossRefGoogle Scholar
  8. [8]
    N. Robertson and P. D. Seymour, Graph Minors III. Planar tree-width, J. Combin. Theory Ser. B, 36 (1984), 49–63.zbMATHCrossRefGoogle Scholar
  9. [9]
    A. Satyanarayana and L. Tung, A characterization of partial 3-trees, Networks, 20 (1990), 299–322.zbMATHCrossRefGoogle Scholar
  10. [10]
    P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B, 28 (1980), 305–359.zbMATHCrossRefGoogle Scholar
  11. [11]
    R. Thomas, Recent excluded minor theorems for graphs, Surveys in Combinatorics, (eds. J. D. Lamb and D. A. Preece), Cambridge University Press, Cambridge, 1999, pp. 201–222.Google Scholar
  12. [12]
    W. T. Tutte, A theory of 3-connected graphs, Indag. Math., 23 (1961), 441–455.Google Scholar
  13. [13]
    K. Wagner, Über eine Eigenschaft der ebenen Komplexe, Math. Ann., 114 (1937), 570–590.zbMATHCrossRefGoogle Scholar
  14. [14]
    K. Wagner, Bemerkungen zu Hadwigers Vermutung, Math. Ann., 141 (1960), 433–451.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Faculty of Mathematics, Physics and InformaticsComenius UniversityBratislavaSlovakia

Personalised recommendations